definition of continuity of a function at x:------------------------------------------------Let f be a function defined on an open interval of x.

Now/then, f is said to be continuous at x if/iff some predicate P(f,x) is true.

is this equivalent to? :

f is said to be continuous at x if/iff f is a function defined on an open interval of x and some predicate P(f,x) is true.

That is, when the first statement is made ahead of the definition proper, how do we interpret it? If some point does not meet the initial statement, are we allowed to call the function "discontinuous" at that point?